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Arithmetic Statistics in Function Fields Arithmetic Statistics in Function Fields

Arithmetic Statistics in Function Fields - PowerPoint Presentation

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Arithmetic Statistics in Function Fields - PPT Presentation

The Divisor function The divisor function counts the number of divisors of an integer Dirichlet divisor problem Determine the asymptotic behaviour as of the sum This is a count of lattice points under the hyperbola ID: 269866

divisor function short intervals function divisor intervals short dirichlet field number points lattice problem fields polynomial polynomials arithmetic line characters rudnick rodgers

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Slide1

Arithmetic Statistics in Function FieldsSlide2

The Divisor function

The

divisor function counts the number of divisors of an integer . Dirichlet divisor problem:Determine the asymptotic behaviour as of the sum This is a count of lattice points under the hyperbola

 Slide3

Counting method

Make use of the symmetry of the region about the line

count twice the number of lattice points under the hyperbola and under the line . * On the diagonal line there are lattice points.* On horizontal lines we have lattice points. Slide4

Therefore

Use

and the Euler summation formula

 Slide5

Dirichlet

‘s divisor problem:

Dirichlet: Voronoi (1903): Huxley (2003):Problem (Divisor function in short intervals): The limiting distribution of when .

The trivial range:

For

and

, as

 Slide6

The Divisor function in short intervals

Define

=Ivic (2009): For with a certain cubic polynomial.  Slide7

The generalized Divisor function

The

k-th divisor function : The classical divisor function being Example: for a prime number , =

Generalization of

Dirichlet

‘s divisor problem:

,

where

is a certain polynomial of degree

 Slide8

The generalized Divisor function in short intervals

Lester

(2015):, then (assuming ) Conjecture J.P.Keating, B.Rodgers, ER-G and Z.Rudnick (2015) ,

,

then

Where

is a piecewise polynomial function in

,

of degree

 Slide9

Function fields

a finite field ( is a power of an odd prime) the ring of polynomials with coefficients in . the set of polynomials of degree be the subset of monic polynomials.The norm of

is

defined by

.

The k-

th

divisor function

Compare with number fields:

 Slide10

Divisors in function fields

The analogue of

Dirichlet divisor problem:Over function fields- an easy computation:The k-th power of the zeta function is the generating function of

Expand + compare the coefficient of

.

 Slide11

Short intervals in

 For and an interval around of length is Note that .The sum over “short interval” The mean value is

Define

Our goal

:

to

study

the variance

of

(in the limit of a large field size)

 Slide12

Short intervals as arithmetic progressions

There is a bijection between Intervals and arithmetic progressions:

When , deg This covers all intervals since  Slide13

Variance in short intervals

Theorem (

J.P.Keating, B.Rodgers, E.R-G and Z.Rudnick 2015) Let and , then as Where

and

are the secular

coefficients:

Corollary:

If

and

, then as

 Slide14

Comparison between number field and function field results (short intervals)

Number field

Function field If then

Number field

Function fieldSlide15

Ingredients of the proof

Orthogonality relation for

Dirichlet characters mod Q: Even characters for all

Riemann Hypothesis (proved by Weil) :

Spectral interpretation (for primitive even characters):

,

 Slide16

The main ingredient

Theorem

(Nick Katz 2013)The unitarized Frobenii when is an even primitive character mod become equidistributed in PU(m-1) as

 Slide17

Sketch of the proof

evaluated in terms of evaluate in terms of the associated Dirichlet L- functionsWrite the L-functions in terms of unitary matrices The variance

 Slide18

Sketch of the proof

Apply Katz

equidistribution result

 Slide19

Matrix Integral

What do we know about

?For , Functional equation

 Slide20

Theorem (

J.P.Keating

, B.Rodgers, ER-G and Z.Rudnick) is equal to the count of lattice points satisfying each of the relations for all

, where

is the collection of

matrices whose entries satisfy the following system of equalities,

 Slide21

Let

. Then for

With

Here

is the Barnes G-function, so that for positive integers k,

is a piecewise polynomial which changes when ever

reaches an integer.

 Slide22

Thank you!